Lemma 12.3 ([1]}) 
Suppose that \(x,y,z\)  are three distinct vertices in \(T={T_{\rm SAW}}(w)\)  such that \(y \)  is on the unique path from \(x\)  to \(z\) . Then
\(\nonumber {{\mathcal {I}}}^{\Gamma ,\sigma }_{T}(x,z)= {{\mathcal {I}}}^{\Gamma ,\sigma }_{T}(x, y)\times {{\mathcal {I}}}^{\Gamma ,\sigma }_{T}(y, z).\) 

